Bianalytic Maps Between Free Spectrahedra

Abstract

Linear matrix inequalities (LMIs) $I_d + \sum_{j=1}^g A_jx_j + \sum_{j=1}^g A_j^*x_j^*\succeq0$ play a role in many areas of applications and the set of solutions to one is called a spectrahedron. LMIs in (dimension--free) matrix variables model most problems in linear systems engineering, and their solution sets D_A are called free spectrahedra. These are exactly the free semialgebraic convex sets. This paper studies free analytic maps between free spectrahedra and, under certain irreducibility assumptions, classifies all those that are bianalytic. The foundation of such maps turns out to be a very small class of birational maps we call convexotonic. The convexotonic maps in g variables sit in correspondence with g-dimensional algebras. If two bounded free spectrahedra D_A and D_B meeting our irreducibility assumptions are free bianalytic with map denoted p, then p must (after possibly an affine linear transform) extend to a convexotonic map corresponding to a g-dimensional algebra spanned by (U-I)A_1,...,(U-I)A_g for some unitary U. Furthermore, B and UA are unitarily equivalent. The article also establishes a Positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from D_A to D_B (not necessarily bianalytic). Another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron. These theorems are needed for classifying free bianalytic maps.